I've made this post as detailed as I can so as to give you a fair idea of my level.
Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I can. Of late, I've been exposed to several problems on Summations, Integration and Special Functions, as well as the knowledge of theorems such as the Dominated Convergence Theorem, Fubini's Theorem and so on so forth. These problems require greater knowledge of Calculus than what I presently know. This has forced me to search for books on Calculus at higher levels so that I can learn more.
Till date, I have learnt Calculus from thousands of sites on the Internet, and from the following 3 books:
Problems in Calculus of One Variable by IA Maron
Mathematical Methods in the Physical Sciences by Mary L Boas
Integral Calculus by Amit M Agarwal
I did as much as I could of Calculus of One Variable a while back and so now have been trying Multi Variable Calculus extensively. I'm quite comfortable (actually done with:)) with Partial Differentiation, Complex Numbers, Trigonometry and to an extent Hyperbolic Trigonometry too. I've done the basics of Multiple Integrals and the basics of some Special Functions (the Beta and Gamma Functions, the Riemann Zeta Function, the Polygamma Functions and the Polylogarithm Functions) too. However, I have never done Linear Algebra or Measure Theory before. I'm also not good at evaluating sums such as Euler Sums.
I would really, really love to to start with Complex methods of Integration such as Contour Integration. However, I've never done it before, so would need a book that starts from scratch.
Under the present circumstances, could you please recommend books which would suitable for me? I'm afraid I tried Apostol, but it was a bit too hard for me. Could somebody suggest books and sources of appropriate standards from where I could study?
I'm most interested in Contour Integration and would really like a book which emphasises it!